What is a pentagonal sphere?
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals).
Why sphere is not a polyhedron?
A polyhedron is a 3-dimensional figure that is formed by polygons that enclose a region in space. Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons. A prism is a polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles.
How many faces does a pentagonal hexecontahedron have?
In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or ” enantiomorphs “) of each other. It has 92 vertices that span 60 pentagonal faces.
Can a hexecontahedron be made from a snub dodecahedron?
The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual. Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon.
How are generalized pentagonal numbers related to centered hexagonal numbers?
Generalized pentagonal numbers are closely related to centered hexagonal numbers. When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper:
How to test if a number is a pentagonal number?
Tests for pentagonal numbers. Given a positive integer x, to test whether it is a (non-generalized) pentagonal number we can compute. The number x is pentagonal if and only if n is a natural number. In that case x is the nth pentagonal number.